## Narrative

Students begin the course with transformational geometry. They study rigid transformations and congruence, then scale drawings, dilations, and similarity (this provides background for understanding the slope of a line in the coordinate plane). Next, they expand their ability to work with linear equations in one and two variables and deepen their understanding of equivalent expressions. They then build on their understanding of proportional relationships from the previous course to study linear relationships. They express linear relationships using equations, tables, and graphs, and make connections across these representations. Building on their understanding of a solution to an equation in one or two variables, they understand what is meant by a solution to a system of equations in two variables. They apply their understanding of linear relationships to contexts involving data with variability. They learn that linear relationships are an example of a special kind of relationship called a function. They extend the definition of exponents to include all integers, and in the process codify the properties of exponents. They learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities. They encounter irrational numbers for the first time and informally extend the rational number system to the real number system, motivated by their work with the Pythagorean Theorem. The last unit offers students an optional opportunity to synthesize their learning from the year using a number of different applications.

### Unit 1: Rigid Transformations and Congruence

Work with transformations of plane figures in this unit draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students’ work with geometric measurement began with length and continued with area. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths. In a previous course, students combined their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra.

Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them (MP1), students use and extend their knowledge of geometry and geometric measurement. They begin the unit by looking at pairs of cartoons, each of which illustrates a translation, rotation, or reflection. Students describe in their own words how to move one cartoon figure onto another. As the unit progresses, they solidify their understanding of these transformations, increase the precision of their descriptions (MP6), and begin to use associated terminology, recognizing what determines each type of transformation, for example, two points determine a translation.

In the first few lessons, students encounter examples of transformations in the plane, without the added structure of a grid or coordinates. The reason for this choice is to avoid limiting students’ schema by showing the least restrictive examples of transformations. Specifically, students see examples of translations in any direction, rotations by any angle, and reflections over any arbitrary line. Through these examples, they begin to understand the features of these transformations without having their understanding limited to, for example, horizontal or vertical translations or rotations only by 90 or 180 degrees. Also, through the use of transparencies, students’ initial understanding of transformations involves moving the entire plane, rather than just moving a given figure. Since all transformations are transformations of the plane, it is preferable for students to first encounter examples that involve moving the entire plane.

They identify and describe translations, rotations, and reflections, and sequences of these. In describing images of figures under rigid transformations on and off square grids and the coordinate plane, students use the terms “corresponding points,” “corresponding sides,” and “image.” Students learn that angles and distances are preserved by any sequence of translations, rotations, and reflections, and that such a sequence is called a “rigid transformation.” They learn the definition of “congruent”: two figures are said to be congruent if there is a rigid transformation that takes one figure to the other. Students experimentally verify the properties of translations, rotations, and reflections, and use these properties to reason about plane figures, understanding informal arguments showing that the alternate interior angles cut by a transversal have the same measure and that the sum of the angles in a triangle is 180 degrees. The latter will be used in a subsequent unit on similarity and dilations. Students end the unit investigating whether sets of angle and side length measurements determine unique triangles (that is, triangles that are congruent), multiple triangles, or fail to determine triangles. Students also study and apply angle relationships, learning to understand and use the terms “complementary,” “supplementary,” “vertical angles,” and “unique” (MP6). Throughout the unit, students discuss their mathematical ideas and respond to the ideas of others (MP3, MP6).

Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts are sometimes contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have opportunities in the unit to tackle real-world applications. In the culminating activity of the unit, students examine and create different patterns formed by plane figures. This is an opportunity for them to apply what they have learned in the unit (MP4).

In this unit, several lesson plans suggest that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to develop their abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools. Apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, generalizing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• movements of figures (Lessons 1 and 2)
• observations about transforming parallel lines (Lesson 8)
• transformations using corresponding points, line segments, and angles (Lesson 9)
• observations about angle measurements (Lesson 14)
• positioning and movement of side lengths and angles (Lesson 15)
• transformations found in tessellations and in designs with rotational symmetry (Lesson 18)

Generalize

• about categories for movement (Lesson 2)
• about rotating line segments $$180^\circ$$ (Lesson 7)
• about the relationship between vertical angles (Lesson 8)
• about corresponding segments and length (Lesson 11)
• about alternate interior angles (Lesson 12)
• about the sum of angles in a triangle (Lesson 14)
• about categories for unique triangles (Lesson 16)

Justify

• whether or not rigid transformations could produce an image (Lesson 6)
• whether or not shapes are congruent (Lesson 10)
• whether or not polygons are congruent (Lesson 11)
• whether or not triangles can be created from given angle measurements (Lesson 13)
• whether or not shapes are identical copies (Lesson 15)
• whether or not measurements determine unique triangles (Lesson 17)

In addition, students are expected to explain and interpret directions for transforming figures and how to apply transformations to find specific images. Students are also asked to use language to compare rotations of a line segment, compare perimeters and areas of rectangles, and compare triangles in a set. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
Acc7.1.1 vertex
plane
measure
direction
slide
turn
Acc7.1.2

clockwise
counterclockwise
reflection
rotation
translation

opposite
Acc7.1.3 image
angle of rotation
center (of rotation)
line of reflection
sequence of transformations
distance

vertex
clockwise
counterclockwise

reflect
rotate
translate

Acc7.1.4 coordinate plane
point
segment
coordinates
$$x$$-axis
$$y$$-axis
Acc7.1.5 polygon angle of rotation
center (of rotation)
line of reflection
Acc7.1.6 rigid transformation
corresponding

measurements
preserve
reflection
rotation
translation

measure
point
Acc7.1.7 midpoint segment
Acc7.1.8 vertical angles
parallel
intersect
distance
Acc7.1.9 image
rigid transformation

midpoint
parallel
Acc7.1.10 congruent
perimeter
area
Acc7.1.11

right angle
corresponding

$$x$$-axis
$$y$$-axis
area
Acc7.1.12 alternate interior angles
transversal
supplementary
complementary
vertical angles
congruent
Acc7.1.13 straight angle supplementary
Acc7.1.14 alternate interior angles
transversal
straight angle
Acc7.1.15 compass
identical copy
condition
different triangle
Acc7.1.16 condition
different triangle
Acc7.1.17 unique triangle protractor
compass
Acc7.1.18 tessellation
symmetry

### Unit 2: Scale Drawings, Similarity, and Slope

Work with scale drawings in this unit draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students’ work with geometric measurement began with length and continued with area. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as an array of unit squares, or rows or columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of a rectangle to include rectangles with fractional side lengths. In grade 6, students built on their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra.

In this unit, students begin with studying scaled copies of pictures and plane figures, then apply what they have learned to scale drawings, such as maps and floor plans. This work leads to making scaled copies using a dilation given a center of dilation and a scale factor. Combining dilations with transformations leads students to defining similarity and determining if two shapes are similar to one another through a series of transformations and a dilation.

Students begin by looking at copies of a picture, some of which are to scale, and some of which are not. They use their own words to describe what differentiates scaled and non-scaled copies of a picture. As the unit progresses, students learn that all lengths in a scaled copy are multiplied by a scale factor and all angles stay the same. They draw scaled copies of figures. They learn that if the scale factor is greater than 1, the copy will be larger, and if the scale factor is less than 1, the copy will be smaller. They study how area changes in scaled copies of an image.

Next, students study scale drawings. They see that the principles and strategies that they used to reason about scaled copies of figures can be used with scale drawings. They interpret and draw maps and floor plans. They work with scales that involve units (for example, “1 cm represents 1 m”), and scales that do not include units (for example, “the scale is 1 to 100”). They learn to express scales with units as scales without units, and vice versa. They understand that actual lengths are products of a scale factor and corresponding lengths in the scale drawing, thus lengths in the drawing are the product of the actual lengths and the reciprocal of that scale factor. They study the relationship between areas and lengths in scale drawings. Throughout the unit, they discuss their mathematical ideas and respond to the ideas of others (MP3, MP6). In the culminating lesson of this unit, students make a floor plan of their classroom or some other room or space at their school. This is an opportunity for them to apply what they have learned in the unit to the world around them (MP4).

Once students understand the ideas of scaled copies limited to pairs of figures with the same rotation and mirror orientation (that is, pairs that are not rotations or reflections of each other), they move on to study pairs of scaled copies that have a different rotation or mirror orientation. They consider cut-out figures, first comparing them visually to determine if they are scaled copies of each other, then representing the figures in a diagram, and then representing them on a circular grid with radial lines. They encounter the new terms “dilation” and “center of dilation.” In the next lesson, students again use a circular grid with radial lines to understand that under a dilation the image of a circle is a circle and the image of a line is a line parallel to the original.

During the rest of the unit, students draw images of figures under dilations on and off square grids and the coordinate plane. In describing correspondences between a figure and its dilation, they use the terms “corresponding points,” “corresponding sides,” and “image.” Students learn that angle measures are preserved under a dilation, but lengths in the image are multiplied by the scale factor. They learn the definition of “similar”: two figures are said to be similar if there is a sequence of translations, rotations, reflections, and dilations that takes one figure to the other. They use the definition of “similar” and properties of similar figures to justify claims of similarity or non-similarity and to reason about similar figures (MP3).

Using these properties, students conclude that if two triangles have two angles in common, then the triangles must be similar. Students also conclude that the quotient of a pair of side lengths in a triangle is equal to the quotient of the corresponding side lengths in a similar triangle. This conclusion is used in the lesson that follows: students learn the terms “slope” and “slope triangle,” and use the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope (MP7). In the following lesson, students use their knowledge of slope to find an equation for a line. They will build on this initial work with slope in a subsequent grade 8 unit on linear relationships. Throughout the unit, students discuss their mathematical ideas and respond to the ideas of others (MP3, MP6).

Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts are sometimes contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have opportunities in the unit to tackle real-world applications. In the culminating activity of the unit, students examine shadows cast by objects in the Sun. This is an opportunity for them to apply what they have learned about similar triangles (MP4).

In this unit, several lesson plans suggest that each student have access to a geometry toolkit. Each toolkit contains tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to develop their abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as explain, describe, and justify. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• how to use scale drawings to find actual distances (Lesson 4 and 7)
• how to apply dilations to find specific images (Lesson 10)
• how to determine whether triangles are congruent, similar, or neither (Lesson 13)
• strategies for finding missing side lengths (Lesson 14)
• how to apply dilations to find specific images of points (Lesson 17)
• reasoning for a conjecture (Lesson 19)

Describe

• features of scaled copies (Lesson 1)
• observations about scaled rectangles (Lesson 8)
• observations about dilated points, circles, and polygons (Lesson 9)
• sequences of transformations (Lesson 11)
• observations about side lengths in similar triangles (Lesson 14)
• relevant features of a classroom with a scale drawing (Lesson 18)

Justify

• reasoning about scaled copies (Lesson 1 and 2)
• that polygons are similar (Lesson 12)
• that triangles are similar (Lesson 15)
• why the height of objects and length of their shadows is approximately proportional (Lesson 19)

In addition, students are expected to generalize about corresponding distances and angles in scaled copies, scale factors, and about points on a line and similar triangles. They represent dilations using given scale factors and graphs of lines using equations. Students are also asked to compare how different scales affect drawings and methods for determining similarity. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
Acc7.2.1 scaled copy
Acc7.2.2 scale factor
Acc7.2.3

scale factor

Acc7.2.4

scale drawing
scale

two-dimensional
three-dimensional

scaled copy
Acc7.2.5 scale
Acc7.2.7 scale without units
equivalent scales
scale drawing
Acc7.2.8 scale factor
scaled copy
scaling
Acc7.2.9 dilation
center of a dilation

dilate
Acc7.2.10 center of a dilation
scale factor
Acc7.2.11 similar dilate
Acc7.2.12 dilation
Acc7.2.14 quotient
Acc7.2.15 similar slope
slope triangle
Acc7.2.16 similarity
$$x$$-coordinate
$$y$$-coordinate
equation of a line
quotient
Acc7.2.19 estimate
approximate / approximately

Teachers Notes for IM 6–8 Math Accelerated
Several supports for English language learners in the first 2 sections of this unit start with a phrase such as “This is the first time Math Language Routine 1 is suggested as a support in this course,” and follow with a detailed description of how the routine is meant to be used in the activity. While the detailed descriptions are still accurate, due to how lessons and units are arranged in IM 6–8 Math Accelerated, these instances may no longer be the first time a specific math language routine is seen.

### Unit 3: Writing and Solving Equations

In this unit, students solve equations of the forms $$px+q=r$$ and $$p(x+q)=r$$ where $$p$$, $$q$$, and $$r$$ are rational numbers.

In the first section of the unit, students represent relationships of two quantities with tape diagrams and with equations, and explain correspondences between the two types of representations (MP1). They begin by examining correspondences between descriptions of situations and tape diagrams, then draw tape diagrams to represent situations in which the variable representing the unknown is specified. Next, they examine correspondences between equations and tape diagrams, then draw tape diagrams to represent equations, noticing that one tape diagram can be described by different (but related) equations. At the end of the section, they draw tape diagrams to represent situations in which the variable representing the unknown is not specified, then match the diagrams with equations. The section concludes with an example of the two main types of situations examined, characterized in terms of whether or not they involve equal parts of an amount or equal and unequal parts of an amount, and as represented by equations of different forms, e.g., $$6(x+8)=72$$ and $$6x+8=72$$. This initiates a focus on seeing two types of structure in the situations, diagrams, and equations of the unit (MP7).

In the second section of the unit, students solve equations of the forms $$px+q=r$$ and $$p(x+q)=r$$, then solve problems that can be represented by such equations (MP2). They begin by considering balanced and unbalanced “hanger diagrams,” matching hanger diagrams with equations, and using the diagrams to understand two algebraic steps in solving equations of the form $$px+q=r$$: subtract the same number from both sides, then divide both sides by the same number. Like a tape diagram, the same balanced hanger diagram can be described by different (but related) equations, e.g., $$3x+6=18$$ and $3(x+2)=18. The second form suggests using the same two algebraic steps to solve the equation, but in reverse order: divide both sides by the same number, then subtract the same number from both sides. Each of these algebraic steps and the associated structure of the equation is illustrated by hanger diagrams (MP1, MP7). So far, the situations in the section have been described by equations in which $$p$$ is a whole number, and $$q$$ and $$r$$ are positive (and frequently whole numbers). In the remainder of the section, students use the algebraic methods that they have learned to solve equations of the forms $$px+q=r$$ and $$p(x+q)=r$$ in which $$p$$, $$q$$, and $$r$$ are rational numbers. They use the distributive property to transform an equation of one form into the other (MP7) and note how such transformations can be used strategically in solving an equation (MP5). They write equations in order to solve problems involving percent increase and decrease (MP2). Progression of Disciplinary Language In this unit, teachers can anticipate students using language for mathematical purposes such as interpret, compare, and explain. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to: Interpret • non-proportional situations with constant rates of change (Lesson 1) • solutions to equations (Lesson 4 and 5) • non-proportional situations with constant rates of change (Lesson 11) • equations representing angle measurements (Lesson 13) Compare • stories with corresponding tape diagrams (Lesson 2) • tape diagrams with corresponding equations (Lesson 3) • hanger diagrams and equations (Lesson 7) • solution pathways (Lesson 10) Explain • strategies for using hanger diagrams to solve equations using hanger diagrams (Lesson 8) • strategies for solving equations (Lesson 9) • reasoning about situations, tape diagrams, and equations (Lesson 12) • how to find unknown angle measurements (Lesson 13) In addition, students are expected to represent non-proportional situations using tape diagrams. They describe the structure of equations and tape diagrams and also generalize about solving equations. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas. The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced. lesson new terminology receptive productive Acc7.3.2 unknown amount Acc7.3.3 equivalent expressions commutative (property) Acc7.3.4 unknown amount relationship Acc7.3.6 variable Acc7.3.7 balanced hanger each side (of an equation) Acc7.3.8 equivalent expression each side (of an equation) Acc7.3.9 operation solve Acc7.3.10 distribute substitute ### Unit 4: Inequalities, Expressions, and Equations In this unit, students expand on their previous work writing and solving equations of the form $$px+q=r$$ and $$p(x+q)=r$$ in three directions: inequalities, equivalent expressions, and solving equations with a variable on each side. They gain greater fluency working with more complicated expressions and refine their understanding about what it means to be a solution to an inequality or equation. The work in this unit leads directly into work in a future unit on linear relationships, and, in particular, systems of equations. In the first section of the unit, students work with inequalities. The first half of the section introduces students to using simple inequalities with variables, such as $$x<r$$ and $$x>r$$. The second half focus on inequalities of the form $$px+q>r$$ or $$px+q<r$$, where $$p$$, $$q$$, and $$r$$ are specific rational numbers, and builds on the work students did in an earlier unit with equations of the same form. Throughout the section, students examine values that make an inequality true or false, and use the number line to represent values that make an inequality true. They also represent situations that involve inequalities, symbolically and with the number line, understanding that there may be infinitely many solutions for an inequality. They solve equations, examine values to the left and right of a solution, and use those values in considering the solution of a related inequality. In the last two lessons of the section, students solve inequalities that represent real-world situations (MP2). In the second section of the unit, students work with equivalent linear expressions, using properties of operations to explain equivalence (MP3). They represent expressions with diagrams, and use the distributive property to justify factoring or expanding an expression. Throughout this section students have opportunities to look for and make use of structure as they combine like terms and apply the distributive property (MP7). The third section focuses on linear equations in one variable. While in previous units students have worked with equations with one occurrence of one variable, now they learn algebraic methods to solve linear equations with multiple occurrences of one variable. They analyze groups of linear equations in one unknown, noting that they fall into three categories: no solution, exactly one solution, and infinitely many solutions. They learn that any one such equation is false, true for one value of the variable, or (using properties of operations) true for all values of the variable. Given descriptions of real-world situations, students write and solve linear equations in one variable, interpreting solutions in the contexts from which the equations arose (MP2). Progression of Disciplinary Language In this unit, teachers can anticipate students using language for mathematical purposes such as justify, critique, and generalize. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to: Justify • reasoning about solutions to inequalities (Lesson 2) • reasoning about solutions to inequalities (Lesson 5) • the need for specific information in order to write and solve inequalities (Lesson 6) • reasoning about the distributive property (Lesson 8) • predictions about solutions of linear equations (Lesson 14) Critique • the reasoning of others (Lesson 3) • reasoning of peers about expressions and corresponding diagrams (Lesson 7) • reasoning about equivalent expressions (Lesson 10) • reasoning about maintaining balance in equations (Lesson 12) • solutions of linear equations (Lesson 13) Generalize • about the relationships between shapes (Lesson 3) • about when expressions are equivalent (Lesson 9) • about the structures of equations that have infinite and no solutions (Lesson 15) • about the structures of equations that have one, infinite, and no solutions (Lesson 16) In addition, students represent inequalities in multiple ways and describe situations involving minimums and maximums. They explain strategies for solving inequalities and for identifying and writing equivalent expressions. Students also compare features of linear equations and their solutions. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas. The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced. lesson new terminology receptive productive Acc7.4.1 less than or equal to greater than or equal to Acc7.4.2 requirement open / closed circle solution to an inequality Acc7.4.3 unbalanced hanger boundary inequality Acc7.4.4 direction (of an inequality) expression less than greater than Acc7.4.5 open / closed circle Acc7.4.6 inequality solution to an inequality Acc7.4.7 term Acc7.7.8 factor (an expression) expand (an expression) Acc7.4.9 combine like terms term commutative (property) expression Acc7.4.10 distribute Acc7.4.11 associative property factor (an expression) expand (an expression) Acc7.4.12 solution to an equation distribute Acc7.4.13 substitute equation Acc7.4.14 term like terms common denominator Acc7.4.15 no solution (only) one solution Acc7.4.16 constant term coefficient linear equation infinitely many solutions variable ### Unit 5: Linear Relationships Work with linear relationships in this unit builds on earlier work with rates and proportional relationships in grade 6, and earlier work with geometry. At the end of the earlier unit on dilations, students learned the terms “slope” and “slope triangle,” used the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope, and found an equation for a line with a positive slope and vertical intercept. In this unit, students gain experience with linear relationships and their representations as graphs, tables, and equations through activities designed and sequenced to allow them to make sense of problems and persevere in solving them (MP1). They revisit earlier work with equations as they study systems of linear equations. The final sections of this unit ask students to examine bivariate data where they use scatter plots and fitted lines to analyze numerical data. The unit begins by revisiting different representations of proportional relationships (graphs, tables, and equations), and the role of the constant of proportionality in each representation and how it may be interpreted in context (MP2). Next, students analyze the relationship between the number of cups in a given stack of cups and the height of the stack—a relationship that is linear but not proportional—in order to answer the question “How many cups are needed to get to a height of 50 cm?” They are not asked to solve this problem in a specific way, giving them an opportunity to choose and use strategically (MP5) representations that appeared earlier in this unit (table, equation, graph) or in earlier units (equation, graph). Students are introduced to “rate of change” as a way to describe the rate per 1 in a linear relationship and note that its numerical value is the same as that of the slope of the line that represents the relationship. Students analyze another linear relationship (height of water in a cylinder vs number of cubes in the cylinder) and establish a way to compute the slope of a line from any two distinct points on the line via repeated reasoning (MP8). They learn a third way to obtain an equation for a linear relationship by viewing the graph of a line in the coordinate plane as the vertical translation of a proportional relationship (MP7). So far, the unit has involved only lines with positive slopes and $$y$$-intercepts. Students next consider the graph of a line with a negative $$y$$-intercept and equations that might represent it. They consider situations represented by linear relationships with negative rates of change, graph these (MP4), and interpret coordinates of points on the graphs in context (MP2). The fourth section focuses on systems of linear equations in two variables. Given descriptions of two linear relationships students interpret points on their graphs, including points on both graphs. Students categorize pairs of linear equations graphed on the same axes, noting that there are three categories: no intersection (lines distinct and parallel, no solution), exactly one intersection (lines not parallel, exactly one solution), and same line (infinitely many solutions). In the fifth section students turn their skills identifying features of linear relationships to analyzing bivariate data. They begin with an investigation of a table of data. Students manipulate the data to look for patterns in the table (MP7), then examine a scatter plot of the same data. This motivates the need to use different representations of the same data to find and analyze any patterns. Throughout this section, students make and examine scatter plots, interpreting points in terms of the quantities represented (MP2). They see examples of how a line can be used to model an association between measurements displayed in a scatter plot and they compare values predicted by a linear model with the actual values given in the scatter plot (MP4). They draw lines to fit data displayed in scatter plots and informally assess how well the line fits by judging the closeness of the data points to the lines (MP4). Students compare scatter plots that show different types of associations (MP7) and learn to identify these types, making connections between the overall shape of a cloud of points and trends in the data represented, for example, a scatter plot of used car price vs. mileage shows a cloud of points that descends from left to right and prices of used cars decrease with increased mileage (MP2). They make connections between the overall shape of a cloud of points, the slope of a fitted line, and trends in the data, for example, “a line fit to the data has a negative slope and the scatter plot shows a negative association between the price of a used car and its mileage.” Outliers are informally identified based on their relative distance from other points in a scatter plot. Students examine scatter plots that show linear and non-linear associations as well as some sets of data that show clustering, describing their differences (MP7). The sixth section is optional and focuses on using two-way tables to analyze categorical data (MP4). Students use a two-way frequency table to create a relative frequency table to examine the percentages represented in each intersection of categories to look for any associations between the categories. Students also examine and create bar and segmented bar graphs to visualize any associations. In this unit, several lesson plans suggest that each student have access to a geometry toolkit. Each toolkit contains tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools. On using the terms ratio, rate, and proportion. In these materials, a quantity is a measurement that is or can be specified by a number and a unit, e.g., 4 oranges, 4 centimeters, “my height in feet,” or “my height” (with the understanding that a unit of measurement will need to be chosen). The term ratio is used to mean an association between two or more quantities and the fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are never called ratios. The fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are identified as “unit rates” for the ratio $$a : b$$. The word “per” is used with students in interpreting a unit rate in context, as in “\$3 per ounce,” and “at the same rate” is used to signify a situation characterized by equivalent ratios.

In grades 6–8, students write rates without abbreviated units, for example as “3 miles per hour” or “3 miles in every 1 hour.” Use of notation for derived units such as $$\frac{\text{mi}}{\text{hr}}$$ waits for high school—except for the special cases of area and volume. Students have worked with area since grade 3 and volume since grade 5. Before grade 6, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation in grade 6, they also use $$\text{ cm}^2$$ and $$\text{ cm}^3$$.

A proportional relationship is a collection of equivalent ratios. In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to $$a$$ to $$b$$, $$a:b$$, and $$\frac{a}{b}$$ as “ratios.”

A proportional relationship between two quantities represented by $$a$$ and $$b$$ is associated with two constants of proportionality: $$\frac{a}{b}$$ and $$\frac{b}{a}$$. Throughout the unit, the convention is if $$a$$ and $$b$$ are represented by columns in a table and the column for $$a$$ is to the left of the column for $$b$$, then $$\frac{b}{a}$$.is the constant of proportionality for the relationship represented by the table.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as represent, interpret, and explain. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Represent

• situations involving proportional relationships (Lesson 1)
• constants of proportionality in different ways (Lesson 2)
• slope using expressions (Lesson 6)
• linear relationships using graphs, tables, equations, and verbal descriptions (Lesson 7)
• situations using negative slopes and slopes of zero (Lesson 8)
• situations by graphing lines and writing equations (Lesson 10)
• situations involving systems of linear equations (Lesson 12, 13, and 16)
• data in organized ways (Lesson 17)
• data using two-way tables, bar graphs, and segmented bar graphs (Lesson 23, 24, and 27)
• situations involving linear relationships (Lesson 25)

Interpret

• situations involving proportional relationships (Lesson 1)
• graphs using different scales (Lesson 2)
• slopes and intercepts of linear graphs (Lesson 5)
• situations using negative slopes and slopes of zero (Lesson 8)
• situations involving systems of linear equations (Lesson 13)
• tables and scatterplots of bivariate data (Lesson 18)
• tables, scatterplots, equations, and situations involving bivariate data (Lesson 19)
• situations involving linear relationships (Lesson 25)

Explain

• how to use a graph to determine information about a linear situation. (Lesson 4)
• how to determine slope from a graph (Lesson 5)
• how slope relates to changes in a situation (Lesson 10)
• how to estimate using available data (Lesson 17)
• how to use tables and scatterplots to make estimates and predictions (Lesson 18)
• the meaning of slope for a situation (Lesson 21)
• how to use lines to show associations, identify outliers, and answer questions (Lesson 22)

In addition, students are expected to generalize about equations and linear relationships, to make predictions about the slope of lines, about the structures of systems of equations, about what makes a line fit a data set well, and about categories for sorting scatter plots. They justify correspondences between different representations, which equations correspond to graphs of horizontal and vertical lines, reasoning about linear relationships, and associations between bivariate data. Students are also asked to describe observations about the equation of a translated line, graphs of systems of linear equations, and features of and trends observed in scatterplots. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
Acc7.5.1 represent
scale
label
constant of proportionality
Acc7.5.2 rate of change
Acc7.5.4 linear relationship
constant rate
slope
Acc7.5.5 vertical intercept
$$y$$-intercept
Acc7.5.6 initial (value of amount) constant rate
Acc7.5.7 relate
Acc7.5.8 horizontal intercept
$$x$$-intercept
Acc7.5.9 constraint

rate of change
vertical intercept

$$y$$-intercept
horizontal line
vertical line

Acc7.5.10 solution to an equation with two variables
variable
combination
set of solutions
Acc7.5.12 ordered pair
Acc7.5.13

system of equations
solution to a system of equations

Acc7.5.14 substitution substitute
no solution
(only) one solution
infinitely many solutions
Acc7.5.15 algebraically
Acc7.5.16 system of equations
substitution
Acc7.5.17 scatter plot
Acc7.5.18 attribute
input
output
numerical data
Acc7.5.19 outlier
predict
overpredict
underpredict
linear model
Acc7.5.20 positive association
negative association

linear association
Acc7.5.21 non-linear association
no association
fitted line
Acc7.5.22 cluster independent variable
dependent variable
positive association
negative association

linear association
Acc7.5.23 categorical data
Acc7.5.24 segmented bar graph
relative frequency
two-way (frequency) table

data display
Acc7.5.27 scatter plot
outlier

cluster

### Unit 6: Functions and Volume

In this unit, students are introduced to the concept of a function as a relationship between “inputs” and “outputs” in which each allowable input determines exactly one output. In the first three sections of the unit, students work with relationships that are familiar from previous grades or units (perimeter formulas, proportional relationships, linear relationships), expressing them as functions. In the remaining three sections of the unit, students learn formulas for volumes of prisms, cylinders, cones, and spheres. Students express functional relationships described by these formulas as equations. They use these relationships to reason about how the volume of a figure changes as another of its measurements changes, transforming algebraic expressions to get the information they need (MP1).

The first section begins with examples of “input–output rules” such as “divide by 3” or “if even, then . . . ; if odd, then . . . ” In these examples, the inputs are (implicitly) numbers, but students note that some inputs are not allowable for some rules, e.g., $$\frac{1}{2}$$ is not even or odd. Next, students work with tables that list pairs of inputs and outputs for rules specified by “input–output diagrams,” noting that a finite list of pairs does not necessarily determine a unique input–output rule (MP6). Students are then introduced to the term “function” as describing a relationship that assigns exactly one output to each allowable input.

In the second section, students connect the terms “independent variable” and “dependent variable” (which they learned in grade 6) with the inputs and outputs of a function. They use equations to express a dependent variable as a function of an independent variable, viewing formulas from earlier grades (e.g., $$C=2\pi r$$), as determining functions. They work with tables, graphs, and equations for functions, learning the convention that the independent variable is generally shown on the horizontal axis. They work with verbal descriptions of a function arising from a real-world situation, identifying tables, equations, and graphs that represent the function (MP1), and interpreting information from these representations in terms of the real-world situation (MP2).

The third section of the unit focuses on linear and piecewise linear functions. Students use linear and piecewise linear functions to model relationships between quantities in real-world situations (MP4), interpreting information from graphs and other representations in terms of the situations (MP2). The lessons on linear functions provide an opportunity for students to coordinate and synthesize their understanding of new and old terms that describe aspects of linear and piecewise functions. In working with proportional relationships in an earlier course, students learned the term “constant of proportionality,” and that any proportional relationship can be represented by an equation of the form $$y=kx$$ where $$k$$ is the constant of proportionality, that its graph lies on a line through the origin that passes through Quadrant I, and that the constant of proportionality indicates the steepness of the line. In an earlier unit, students were introduced to “rate of change” as a way to describe the rate per 1 in a linear relationship and noted that its numerical value is the same as that of the slope of the line that represents the relationship. In this section, students connect their understanding of “increasing” and “decreasing” from the previous section with their understanding of linear functions, noting, for example, that if a linear function is increasing, then its graph has positive slope, and that its rate of change is positive. Similarly, they connect their understanding of $$y$$-intercept (learned in an earlier unit) with the new term “initial value,” noting, for example, when the numerical part of an initial value of a function is given by the $$y$$-intercept of its graph (MP1).

In the latter half of the unit, students turn to focus on features of 3 dimensional shapes, using abilities developed in earlier work with geometry and geometric measurement, and consider volume formulas as examples of functions.

Students’ work with geometry began in kindergarten, where they identified and described two- and three-dimensional shapes that included cones, cylinders, and spheres. They continued this work in grades 1 and 2, composing, decomposing, and identifying two- and three-dimensional shapes.

Students’ work with geometric measurement began with length and continued with area and volume. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area models to represent instances of the distributive property. In grade 4, students used area and perimeter formulas for rectangles to solve real-world and mathematical problems. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths. They found volumes of right rectangular prisms by viewing them as layers of arrays of cubes and used formulas to calculate these volumes as products of edge lengths or as products of base area and height. In grade 6, students extended the formula for the volume of a right rectangular prism to right rectangular prisms with fractional side lengths and used it to solve problems. They extended their reasoning about area to include shapes not composed of rectangles and combined their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra.

In this unit, students analyze and describe cross-sections of prisms (including prisms with polygonal but non-rectangular bases), pyramids, and polyhedra. They investigate the volume of water in a graduated cylinder as a function of the height of the water, and vice versa. They estimate volumes of prisms, cylinders, cones, and spheres in order to reinforce the idea that a measurement of volume indicates the amount of space within an object. Students then learn the formula for the volume of a right rectangular prism, and solve problems involving the surface area and volume of prisms. Next, they extend their understanding from right rectangular prisms to right cylinders by perceiving similar structure (MP7) in formulas for the volume of a rectangular prism and the volume of a cylinder—both are the product of base and height. After gaining familiarity with a formula for the volume of a cylinder by using it to solve problems, students perceive similar structure (MP7) in a formula for the volume of a cone.

The fifth section of the unit begins with an examination of functional relationships between two quantities that are illustrated by changes in scale for three-dimensional figures. For example, if the radius of a cylinder triples, its volume becomes nine times larger. This work combines work done in a previous course on scale and proportional relationships. In that course, students studied scaled copies of two-dimensional figures, recognizing lengths are scaled by a scale factor and areas by the square of the scale factor, and applied their knowledge to scale drawings—for example, maps and floor plans. In their previous study of proportional relationships, students solved problems set in contexts commonly associated with proportional relationships such as constant speed, unit pricing, and measurement conversions, and learned that any proportional relationship can be represented by an equation of the form $$y=kx$$ where $$k$$ is the constant of proportionality. In this section, students use their knowledge of scale, proportional relationships, and volume to reason about how the volume of a prism, cone, or cylinder changes as another measurement changes.

In the following section, students reason about how the volume of a sphere changes as its radius changes. They consider a situation in which water flows into a cylinder, cone, and sphere at the same constant rate. Information about the height of the water in each container is shown in an equation, graph, or table, allowing students to use it strategically (MP5) to compare water heights and capacities for the containers.

Progression of Disciplinary Language
In this unit, teachers can anticipate students using language for mathematical purposes such as compare, explain, and describe. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Compare

• different representations of functions. (Lesson 3)
• features of graphs, equations, and situations. (Lesson 4)
• features of a situation with features of a graph (Lesson 6)
• temperatures shown on a graph with different temperatures given in a table (Lesson 7)
• cross sections of figures (Lesson 11)
• the volumes of cones with the volumes of cylinders (Lesson 16)
• methods for finding and approximating the volume of a sphere as function of its radius (Lesson 24)
• how to determine characteristics of triangles and prisms (Lesson 26)

Explain

• how height and volume of cylinders are related (Lesson 12)
• how to find the area and volume of prisms (Lesson 14 and 15)
• how to find the surface area of prisms (Lesson 16)
• reasoning about finding the volume of a cylinder (Lesson 18)
• reasoning about the relationship between volumes of hemispheres and volumes of boxes, cylinders, and cones (Lesson 23)
• how to determine characteristics of triangles and prisms (Lesson 26)

Describe

• quantities in a situation (Lesson 6)
• approximately linear relationships (Lesson 9)
• relationships that are approximately piecewise linear (Lesson 10)
• cross sections of prisms and pyramids (Lesson 11)
• volume measurements and features of three dimensional figures (Lesson 13)
• the effects of varying dimensions of rectangular prisms and cones on their volumes (Lesson 22)
• how to determine characteristics of triangles and prisms (Lesson 26)

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
Acc7.6.1 input
output
Acc7.6.2 function input
output
depends on
Acc7.6.3 independent variable
dependent variable

Acc7.6.5 prediction
Acc7.6.7 volume
cube
Acc7.6.8 functional relationship
linear function
function
Acc7.6.9 mathematical model prediction
Acc7.6.10 piecewise linear function linear function
constant rate
Acc7.6.11 cross section
base (of a prism or pyramid)

vertex (of a pyramid)
face
prism
pyramid
perpendicular
parallel
Acc7.6.12 cylinder
three-dimensional
Acc7.6.13 cone
sphere
dimension
cylinder
cube
cubic centimeter
rectangular prism
Acc7.6.14 volume
cross section
base (of a prism or pyramid)
Acc7.6.16 face
perimeter
surface area
Acc7.6.18 base (of a cylinder or cone)
approximation for
Acc7.6.20 cone
base (of a cylinder or cone)
Acc7.6.23 hemisphere
Acc7.6.24 sphere
Acc7.6.25 spherical
Acc7.6.27 approximate range

### Unit 7: Exponents and Scientific Notation

Students were introduced to exponent notation in grade 6. They worked with expressions that included parentheses and positive whole-number exponents with whole-number, fraction, decimal, or variable bases, using properties of exponents strategically, but did not formulate rules for use of exponents.

In this unit, students build on their grade 6 work. The first section of the unit begins with a lesson that reviews exponential expressions, including work with exponential expressions with bases 2 and $$\frac 12$$. In the next two lessons, students examine powers of 10, formulating the rules $$10^n \boldcdot 10^m = 10^{n+m}$$, $$({10^n})^m = 10^{n \boldcdot m}$$, and, for $$n>m$$, $$\frac{10^n}{10^m} = 10^{n-m}$$ where $$n$$ and $$m$$ are positive integers. After working with these powers of 10, they consider what the value of $$10^0$$ should be and define $$10^0$$ to be 1. In the next lesson, students consider what happens when the exponent rules are used on exponential expressions with base 10 and negative integer exponents and define $$10^{-n}$$ to be $$\frac{1}{10^n}$$. In the next two lessons, they expand their work to numerical bases other than 10, using exponent rules with products of exponentials with the same base and contrasting it with products of exponentials with different bases. They note numerical instances of $$a^n \boldcdot b^n = (a \boldcdot b)^n$$.

The second section of the unit returns to powers of 10 as a prelude to the introduction of scientific notation. Students consider differences in magnitude of powers of 10 and use powers of 10 and multiples of powers of 10 to describe magnitudes of quantities, for example, the distance from Earth to the Sun or the population of Russia. Initially, they work with large quantities, locating powers of 10 and positive-integer multiples of powers of 10 on the number line. Most of these multiples are products of single-digit numbers and powers of 10. The remainder are products of two- or three-digit numbers and powers of 10, allowing students to notice that these numbers may be expressed in different ways, for example, $$75 \boldcdot 10^5$$ can be written $$7.5 \boldcdot 10^6$$, and that some forms may be more helpful in finding locations on the number line. In the next lesson, students do similar work with small quantities.

In the remaining five lessons, students write estimates of quantities in terms of integer or non-integer multiples of powers of 10 and use their knowledge of exponential expressions to solve problems, for example, How many meter sticks does it take to equal the mass of the Moon? They are introduced to the term “scientific notation,” practice distinguishing scientific from other notation, and use scientific notation (with no more than three significant figures) in order to make additive and multiplicative comparisons of pairs of quantities. They compute sums, differences, products, and quotients of numbers written in scientific notation (some with as many as four significant figures), using such calculations to estimate quantities. They make measurement conversions that involve powers of ten, for example, converting bytes to kilobytes or gigabytes, choose appropriate units for measurements and express them in scientific notation.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as critiquing, representing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• reasoning about powers of powers (Lesson 3)
• reasoning about zero exponents (Lesson 4)
• applications of exponent rules (Lesson 7)
• reasoning about scientific notation (Lesson 14)

Represent

• situations using exponents (Lesson 1)
• large and small numbers using number lines, exponents, and decimals (Lesson 9–11)
• situations comparing quantities expressed in scientific notation (Lesson 13)

Justify

• reasoning about multiplying powers of 10 (Lesson 2)
• reasoning about powers of powers (Lesson 3)
• reasoning about dividing powers of 10 (Lesson 4)
• whether or not expressions are equivalent to exponential expressions (Lesson 6)
• reasoning about situations comparing powers of 10 (Lesson 12)

In addition, students are expected to use language to generalize reasoning about repeated multiplication and generalize about patterns when multiplying different bases and exponents; describe how negative powers of 10 affect placement of decimals; and interpret situations comparing quantities expressed in scientific notation. Students also have opportunities to compare correspondences between exponential expressions and base-ten diagrams; compare expressions in scientific notation to other expressions; explain how to simplify expressions with negative powers of 10; and explain how to place and order large numbers on a number line.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students&rsqup; use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.7.1 exponent
power
factor
reciprocal
repeated multiplication
8.7.2 powers of 10
8.7.3 base (of an exponent)
power of powers
8.7.4 expanded
positive exponent
zero exponent
8.7.5 negative exponent positive exponent
8.7.6 exponent
base (of an exponent)
power
zero exponent
8.7.7 reciprocal
evaluate
factor
power of powers
negative exponent
8.7.8 square (of a number)
8.7.9 billion
trillion
multiple of
8.7.10 integer
8.7.12 scientific notation multiple of
integer
8.7.13 powers of 10
billion
trillion
8.7.14 scientific notation

### Unit 8: Pythagorean Theorem and Irrational Numbers

Work in this unit is designed to build on and connect students’ understanding of geometry and numerical expressions. The unit begins by foreshadowing algebraic and geometric aspects of the Pythagorean Theorem and strategies for proving it. Students are shown three squares and asked to compare the area of the largest square with the sum of the areas of the other two squares. The comparison can be done by counting grid squares and comparing the counts—when the three squares are on a grid with their sides on grid lines and vertices on intersections of grid lines—using the understanding of area measurement established in grade 3. The comparison can also be done by showing that there is a shape that can be decomposed and rearranged to form the largest square or the two smallest squares. Students are provided with opportunities to use and discuss both strategies.

In the second section, students work with figures shown on grids, using the grids to estimate lengths and areas in terms of grid units, e.g., estimating the side lengths of a square, squaring their estimates, and comparing them with estimates made by counting grid squares. The term “square root” is introduced as a way to describe the relationship between the side length and area of a square (measured in units and square units, respectively), along with the notation $$\sqrt{}$$. Students continue to work with side lengths and areas of squares. They learn and use definitions for “rational number” and “irrational number.” They plot rational numbers and square roots on the number line. They use the meaning of “square root,” understanding that if a given number $$p$$ is the square root of $$n$$, then $$x^2 = n$$. Students learn (without proof) that $$\sqrt 2$$ is irrational. They understand two proofs of the Pythagorean Theorem—an algebraic proof that involves manipulation of two expressions for the same area and a geometric proof that involves decomposing and rearranging two squares. They use the Pythagorean Theorem in two and three dimensions, e.g., to determine lengths of diagonals of rectangles and right rectangular prisms and to estimate distances between points in the coordinate plane.

In the third section, students work with edge lengths and volumes of cubes and other rectangular prisms. (In this grade, all prisms are right prisms.) They are introduced to the term “cube root” and the notation $$\sqrt{}$$. They plot square and cube roots on the number line, using the meanings of “square root” and “cube root,” e.g., understanding that if a given number $$x$$ is the square root of $$n$$ and $$n$$ is between $$m$$ and $$p$$, then $$x^2$$ is between $$m$$ and $$p$$ and that $$x$$ is between $$\sqrt{m}$$ and $$\sqrt{p}$$.

In the fourth section, students work with decimal representations of rational numbers and decimal approximations of irrational numbers. In grade 7, they used long division in order to write fractions as decimals and learned that such decimals either repeat or terminate. They build on their understanding of decimals to make successive decimal approximations of $$\sqrt 2$$ and $$\pi$$ which they plot on number lines.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as explaining, representing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• strategies for finding area (Lesson 1)
• strategies for approximating and finding square roots (Lesson 4)
• strategies for finding triangle side lengths (Lesson 6)
• predictions about situations involving right triangles and strategies to verify (Lesson 10)
• strategies for finding distances between points on a coordinate plane (Lesson 11)
• strategies for approximating the value of cube roots (Lesson 13)

Represent

• square roots on a number line (Lesson 4)
• relationships between side lengths and areas (Lesson 6)
• cube edge lengths and volumes using cube roots (Lesson 10)
• cube roots on a number line (Lesson 10)
• rational numbers as ratios, decimal expansions, and on a number line (Lesson 11)

Justify

• which squares have side lengths in a given range (Lesson 1)
• ordering of irrational numbers (Lesson 5)
• ordering of hypotenuse lengths (Lesson 9)

In addition, students are expected to use language to generalize about area of squares, square roots, and approximations of side lengths and generalize about the distance between any two coordinate pairs; critique reasoning about square root approximations and critique a strategy to represent repeating decimal expansions as fractions; describe observations about the relationships between triangle side lengths and describe hypotenuses and side lengths for given triangles; interpret diagrams involving squares and right triangles; interpret equations and approximations for the value of square and cube roots; compare rational and irrational numbers and strategies for approximating irrational numbers.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
Acc7.8.2 square root square (of a number)
Acc7.8.3 irrational number
square root symbol
rational number
Acc7.8.4 diagonal
decimal approximation
square root
square root symbol
Acc7.8.5 Pythagorean Theorem
hypotenuse
legs
right triangle
Acc7.8.7 converse of the Pythagorean Theorem Pythagorean Theorem
Acc7.8.8 edge length hypotenuse
legs
Acc7.8.10 cube root edge length
Acc7.8.11 repeating decimal
decimal representation
finite decimal expansion
cube root
Acc7.8.12 infinite decimal expansion irrational number
repeating decimal

### Unit 9: Putting It All Together

In this optional unit, students use concepts and skills from previous units to solve four groups of problems. In the first group of lessons, they consider tessellations of the plane, understanding and using the terms “tessellation” and “regular tessellation” in their work, and using properties of shapes (for example, the sum of the interior angles of a quadrilateral is 360 degrees) to make inferences about regular tessellations. In the next group of lessons, students calculate and estimate quantities associated with running a restaurant. They use their knowledge of proportional relationships, survey findings, and scale drawings to determine the number of calories in one serving of a recipe, the expected number of customers served per day, or the floor space needed in the restaurant. Students then estimate quantities such as age in hours and minutes or number of times their hearts have beaten, using measurement conversions and consider the accuracy of their estimates. In the last group of lessons, they investigate relationships of temperature and latitude, climate, season, cloud cover, or time of day. In particular, they use scatter plots and lines of best fit to investigate the question of modeling temperature as a function of latitude.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, representing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• tessellations (Lesson 1)
• associations in bivariate data (Lesson 11)

Represent

• costs of ingredients in a spreadsheet (Lesson 5)
• situations using expressions and equations (Lesson 9)
• the relationship between latitude and weather (Lesson 11)

Justify

• claims about shapes that can and cannot be used to produce regular tessellations (Lesson 2)
• reasoning about the nutritional value of recipes (Lesson 4)
• choices and predictions in the context of running a restaurant (Lesson 6)

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
Acc7.9.1 tessellation
pattern
Acc7.9.2 regular tessellation regular polygon
Acc7.9.3 serving